Incoherent Energy Transfer

Another way that a molecule (the =%9%*, D) can return to the ground state after excitation is to transfer its excitation energy to another molecule (the

?&&"/$%*, A) which is thereby raised to a higher energy state:

* *

D 0A^NNOD A0 (1.69)

The excitation may in principle be electronic, vibrational, rotational, or translational. The discussion in this work will be restricted to "-"&$*%9(&

"9"*.+ $*?9)!"*, also called "7&($?$(%9 "9"*.+ $*?9)!"* (EET). Energy transfer plays a particularly important role in photosynthetic systems, as found in plants and some bacteria. Sunlight gets absorbed by arrays of pigment-protein complexes which form antenna and light-harvesting sys-tems and subsequently transfer the energy to reaction centres, where it is trapped in electron-transfer reactions [60].

Two main energy transfer mechanisms exist. In the first one, the

*?=(?$(8" energy transfer, also known as trivial mechanism, the excitation energy is transferred by radiative deactivation of a donor molecule and

reabsorption of the emitted radiation by an acceptor molecule [61]. The probability of transfer depends on the overlap between the donor fluores-cence and the acceptor emission spectra, on the acceptor concentration, and on the sample thickness, but not on the distance between donor and acceptor molecules. Radiative transfer results in a decrease of the donor fluorescence intensity in the region of the spectral overlap. This distortion of the fluorescence spectrum is called inner-filter effect. The radiative mechanism has no relevance for the work described hereafter and will not be further commented on.

The other class of mechanisms regroups all 9%9-*?=(?$(8" forms of energy transfer. The process can be divided into four elementary steps:

(1) absorption of a photon by the donor, (2) relaxation of the donor and of its surroundings, (3) energy transfer from the donor to the acceptor, and (4) relaxation of the donor, the acceptor, and the surroundings to adapt to their new states. Step 1 occurs within one oscillation period of the light (up to a few femtoseconds in the visible); steps 2 and 4 typically occur on a 0.1–10 ps time scale in solution; step 3 strongly depends on the nature of the donor and acceptor molecules, on the distance between them, and on the acceptor concentration. If the energy transfer rate is faster than the relaxation steps, back energy transfer can occur, the elec-tronic excitation is delocalised over the donor-acceptor system, and one talks about &%'"*"9$"9"*.+ $*?9)!"*. Such situation is commonly found in molecular organic lattices and dye aggregates which present Frenkel exciton states. This case is discussed in more details in section 1.4.3. If, on the other hand, relaxation dynamics is faster than the energy transfer process, donor and acceptor molecules are in thermal equilibrium with their environment and all coherence is lost. The energy stays localised on either the donor or the acceptor molecule. The process is called (9&%'"*"9$ electronic energy transfer or, if the partners are identical, elec-tronic energy '%//(9.. It typically takes place in dilute solutions or in inorganic matrices containing low concentrations of a doping species.

Incoherent energy transfer mechanisms are now discussed.

Incoherent energy transfer encompasses situations in which the system is at thermal equilibrium with its environment before energy transfer takes place. In this sense, the energy transfer process is irreversible because recurrence of the system to its original situation (configuration of the

do-nor, of the acceptor, and of their surroundings) is not possible.¹¹ Because an irreversible non-radiative transition is considered, the Fermi golden rule (see section 1.2.3) can be applied to evaluate the rate constant of the incoherent energy transfer process 5EET:

2

EET DA DA

5 "2' D 5

# . (1.70)

The coupling DDA between the donor and the acceptor is assumed to be small. This term is the sum of two contributions which are described in the rest of this section: one resulting from a dipole-dipole interaction, Ddip, which is the most commonly used to evaluate 5EET, and another from an electron exchange interaction, Dex, which is often neglected:

DA dip ex

D "D 0D . (1.71)

Accordingly, the energy transfer rate constant can be rewritten as the sum of the rate constants arising from the two contributions, 5dip and 5ex re-spectively:

EET dip ex

5 "5 05 . (1.72)

a) The Dipole-Dipole Approximation: the Förster Mechanism

Theodor Förster was the first to develop a theory to calculate the transi-tion rate of incoherent energy transfer [62, 63]. Within the framework of time-dependent perturbation theory, and given the assumptions that the system be at thermal equilibrium and the coupling between acceptor and donor be small, he considered that the coupling element DDA was mainly due to electrostatic interaction between the donor and the acceptor. This is true if the molecules are sufficiently far apart so that there is no orbital overlap between them and that electrons can unambiguously be assigned to one molecule or to the other. DDA can then be obtained by breaking the electrostatic interaction into monopole-monopole, monopole-dipole,

11 In this case, irreversibility is understood as the loss of coherence, that is, the loss of memory of the initial configuration. Irreversibility does not preclude back energy transfer from the acceptor to the donor, but back energy transfer has to be considered as a new process since it leads to a previously unknown situation of the system.

and dipole-dipole terms. If the molecules have no net charges and the distance between them is large relative to molecular dimensions, the main contribution to DDA usually comes from dipole-dipole interactions and can be expressed as the energy of interaction between two electric “point”

dipoles, Ddip, located at the centres of the two chromophores [13, 64]:

#

D*D DA

$#

AA* DA

$

Because the orientation factor P in equation (1.75) can vary from –2 to +2 depending on the relative orientations of +!_D*D

, +!_AA*

, and *!DA

, D_DA can be either positive or negative. Figure 1.7 illustrates the limiting cases. The sign of P has however no influence on the rate constant since the latter depends on DDA² and thus on P². When the chromophores are ran-domly oriented, as it is the case in an isotropic solution for example, P² has an average value of 2/3 [13].

Corrections to equations (1.73) and (1.74) are needed if the dipoles are embedded in a dielectric medium, as it is the case in solution. For dipoles which fluctuate in position or orientation very rapidly relative to

Figure 1.7. Effect of the relative orientation (parallel, collinear, or perpendicular) of the transition dipole moments on the orientation factor P.

the time scale of nuclear motions, the field in a homogeneous medium of refractive index 9 is reduced by a factor ,op^!1, where ,op C9² is the dielec-tric constant at high (optical) frequencies [13]. Furthermore, viewing the interacting dipoles as embedded in small cavities corresponding roughly to the molecular volumes, a local-field correction factor has to be taken into account for each dipole. Generally, the molecular polarisability is indeed different from the polarisability of the medium, so that the electric fields “inside” the molecule and in the medium will be different. The most commonly used correction factors are the cavity-field and the Lor-entz correction factors [13], the latter one, !L, being defined as

2

It follows that the overall correction to the interaction energy is approxi-mately of !_L²/9² and the expression for DDA becomes

If the coupling energy is expressed in cm^!1, the transition dipoles in units of debyes and the distance in nm, equation (1.77) can be reformulated as [60] constant, one has to take into account the fact that real electronic transi-tions are not sharp lines, but broad bands. The rate for each possible transition has to be calculated and all the contributions are summed up to obtain 5EET. This can be done by combining equations (1.70) and (1.77), by introducing a dependence of the transition dipole moments on energy, 2, and by integrating over it:

# $ # $

be-tween the acceptor absorption spectrum and the area-normalised donor emission spectrum; +^!AA*

# $

2 ² is related to the oscillator strength of the transition and thus to the molar extinction coefficient ^,

# $

² through equa-tion (1.18); +!_D*D²

is connected through the Einstein > coefficient to the radiative lifetime (see equation (1.27)). With these considerations, För-ster’s expression for the rate of resonance energy transfer is obtained, with the distance between the dipoles given in angstroms:

4 2

A more convenient way of calculating the energy transfer rate constant is given by [65] overlap intergral between the area-normalised spectra, ^QR, being defined as

Förster-type energy transfer is restricted to dipole-allowed transitions and its rate also depends on the Franck-Condon factors for the pairs of vertical downward and upward vibronic transitions that satisfy the reso-nance condition. This is the same condition as for optical absorption.

However, although 5EH is estimated from the overlap of the acceptor absorption and donor emission spectra, resonance energy transfer does 9%$ involve emission and reabsorption of a photon, and the common jargon “fluorescence resonance energy transfer” (FRET) is misleading.

The same acronym shall however be used more precisely for “Förster resonance energy transfer” [61]. One should indeed rather think of FRET as the relaxation of the donor and simultaneous excitation of the acceptor through the exchange of a (virtual) photon (Figure 1.8).

The dependence of the energy transfer rate on the sixth power of the intermolecular distance makes FRET a particularly sensitive and useful tool for measuring distances in rigid or semi-rigid systems such as biologi-cal macromolecules, and “molecular rulers” have been developed based on this principle [66-68]. By choosing the appropriate pairs of chromo-phores, energy transfer rates can be measured over distances typically ranging from 10 to 100 Å. The rate for a given donor-acceptor pair can be sensitive enough to afford a resolution on the order of 1 Å [13].

b) The Exchange Interaction: the Dexter Mechanism

Förster theory describes the energy transfer rate in the case of two chro-mophores that lie relatively far apart. It assumes that the dipoles can be assimilated to points and that the intermolecular interactions have no effect on the absorption or fluorescence spectra of the molecules. An-other limitation is that the Förster treatment considers only the spatial Figure 1.8. Schematic representation of the excitation energy transfer through the dipole-dipole interaction leading to simultaneous relaxation of the excited donor and excitation of the acceptor.

parts of the molecular wavefunctions, the spin part being implicitly as-sumed as constant. The theory is therefore not applicable to describe energy transfer processes in which the molecules make transitions be-tween a singlet and a triplet state. To overcome these restrictions, Dexter proposed in 1953 to consider an additional term in the expression for the coupling term in the Fermi golden rule (equation (1.70)) [69]. This term, Dex (see equation (1.71)), takes into account the "-"&$*%9 "7&'?9." be-tween the donor and the acceptor. The rate constant, 5ex, due to this en-ergy transfer mechanism can be defined as

2

ex ex DA

5 " 2' D 5

# . (1.84)

Considering that the density of states, 5_DA, can be calculated in the same way as in the Förster theory and ensures energy conservation, the rate constant for exchange energy transfer, 5_ex, is rewritten in the form

2

ex 2 ex

5 _" ' D _QR

# . (1.85)

The exchange mechanism can be thought of as two simultaneous elec-tron transfer processes, with one elecelec-tron being transferred from the LUMO of the donor to the LUMO of the acceptor, and another electron from the HOMO of the acceptor to the HOMO of the donor (Figure 1.9). Electron exchange requires spatial overlap between the involved molecular orbitals and therefore occurs only at short distances. The abso-lute magnitude of the exchange coupling term is hard to estimate, but since the molecular orbitals fall off exponentially, one may write:

Figure 1.9. Schematic representation of the excitation energy transfer through the exchange interaction leading to simultaneous electron transfer.

# $

magnitude of the coupling at zero distance, and S is a constant describing the distance dependence of the considered orbitals.

Since the exchange interaction decreases much faster with distance than the dipole-dipole interaction, the latter is usually the dominant en-ergy transfer mechanism when the intermolecular distance is larger than about 5 Å and the exchange part can be neglected. There are nonetheless some cases where the exchange mechanism becomes important [64]:

1. When the donor and acceptor parts are linked by a bridge that has empty orbitals close in energy to the HOMO of the donor and that of the acceptor. These orbitals will interact with the donor and acceptor orbitals, which may lead to very low S values and thus to efficient en-ergy transfer over long distances [70, 71].

2. When the transition is not dipole-allowed.

3. When during the energy transfer process both molecules change spin, so that the overall spin quantum number is conserved, as in the reac-tion

3D*⁰1A^NNO1D⁰3A*. (1.87)

Since the Förster mechanism is in such case spin-forbidden, the ex-change mechanism will dominate.

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